Posted
by
timothy
on Monday December 06, 2010 @02:02AM
from the it's-all-mathy dept.

parallel_prankster writes "I find this paper very amusing. From the abstract: 'To develop a mathematical model for the determination of total areas under curves from various metabolic studies.' Hint! If you replace phrases like 'curves from metabolic studies' with just 'curves,' then you'll note that Dr. Tai rediscovered the rectangle method of approximating an integral. (Actually, Dr. Tai rediscovered the trapezoidal rule.). Apparently this is called 'Tai's Model.'"

Really it should be under idle, it's just the fact that the dude forgot all about calculus and went back and remade the approximate method of integration. His hubris must be punished by way of an Internet meme.

Could probably do something with Tai meaning "Red Snapper" in Japanese, or "Wife" in Chinese, but that might be a bit too highbrow for an internet meme.

In any event, it's not hubris to get excited about something you invented that you didn't know existed before. It's ignorance. I once explained to a CS professor this method I'd found for finding the greatest common divisor of two integers, and he cut me off by saying that Euclid had figured it out 2300 years ago.:p

Maybe some bits can and should be taught that way, but the body of knowledge in mathematics is too large to try and teach any significant portion that way. It's taken humanity many lifetimes to discover what we know, one person doesn't have that long. Rediscovering something can be really cool on a one off basis, but there isn't time to do that for the entire body of knowledge nor should we try. Discovery is about the need to know and understand and the drive to sate that need. It's hard to teach those q

> Rediscovering something can be really cool on a one off basis, but there isn't time to do that for the entire body of knowledge nor should we try.

I don't think anyone is arguing to try to teach the WHOLE domain of one field that way. We're talking about the _basics_. What is taught for today's Math is a total joke - kids aren't taught to think, just to mindless follow some "arcane formula". e.g. "Two weeks of content are stretched to semester length by masturbatory definitional runarounds." EVERYONE should read these two papers.

In any event, it's not hubris to get excited about something you invented that you didn't know existed before. It's ignorance.

The two are not mutually exclusive. Going so far as to publish a paper describing something he is expected to have learned in high school or at least in college is over the top.Its pretty bad that the peer review didn't catch it either...

Tai: So as you can see I used this method to calculate the surface underneath the graph, and as you can clearly see the results show that....
Fellow MD1: Wait, what method? That looks pretty sciency!
Fellow MD2: Cool method, did you think of it yourself?
Tai: Huh, I just calculated the surface underneath the graph, it's basic calculus you know?
Fellow MD1: Calculus schmalculus, did you think of publishing your method
Fellow MD2: Yeah you should totally publis

The sad thing is, he probably had some level of Calculus in school, and probably memorized the formulas and rules, did the math and answered the questions and got a passing grade, all without understanding. Education doesn't require understanding, it is just indoctrination of scholarly principles.

Don't blame him for the error, blame the system that allowed it to happen.

"The glucose and insulin responses to the OGTT were analyzed by calculating the area under the curve (AUC). The AUCs for glucose (AUCglucose) and insulin (AUCinsulin) were determined according to the Tai procedure for the metabolic curves (25)."

I wonder if this is sort of an inside joke now. Rather than saying we used the trapezoidal rule to approximate XYZ, everyone in the field now says "we used the Tai procedure". It sounds so much more 'official'. Remind me to reinvent the central limit theorem tomorrow.

And this doesn't help the people trying to fight the stigma that biology isn't a 'hard science'.

And this doesn't help the people trying to fight the stigma that biology isn't a 'hard science'.

The problem isn't that biology isn't a "hard science" (although some branches are pretty soft), the problem is that most MDs aren't real scientists. Ask any biology grad student what it's like to teach pre-meds and you'll get an earful. It's difficult for me to take the profession seriously any more; my employer and I combined are paying $700 per month in case I get sick and need to be treated by some overpaid

You do not need to see the bullets. That's impossible. Instead only try to realize the truth. There are no bullets. Then you'll see, it's not the bullets that need to move, it is only the idea of where they are aren't.

you assume he knew calculus before he started. In terms of relevance to us today, I see this kind of thing all the time in computing - why bother using the standard mechanism of performing a task when tyou can reinvent the wheel all over again. From the innumerable number of programming languages, to open source projects, to just my co-worker making up his own string class (gah!!)

Sometimes I wonder if its a lack of education (or more likely experience), or just bone-headed stubborness to understand anything

That's the difference between software "engineering" and any other form of engineering. Maybe in another 200 years programmers will be there, civil and electrical disciplines have had a fair head start.

in a word, yes, check out almost any medical stats methodology - it looks sort of right if you have only degree level maths but, eg, statisticians have pretty much given up on pointing out that treating binned averages of a population as raw data typically invalidates the method under consideration, rendering the results speculative at best.

researchers will tend to insist that what they have handed over is raw data because they have (or a research associate, or Excel! has) only performed a few simple transformations on it and, that being many months ago, probably have forgotten the fact. one can either keep performing extra (unpaid and unasked for) analyses showing that this distribution verges on the impossible (and risk not be asked for help in future) or shut up and get cited and allow your reputation to grow

having said that, the same is true for many scientific practitioners and, indeed, the majority of published journal papers - the peer review generally doesn't extend to a competent mathematical practitioner (still less frequently a statistician) and most academics do not appear to consider that anything beyond their (often high school- or graduate-level) understanding of mathematics is required, after all (like the paper concerned here) building on previously published and highly cited work of little worth is all that's required for a career

I would skim my girlfriend's Journal of the American Medical Association (JAMA) magazines occasionally and the studies people did in the same of science were appalling.

They'd make medical conclusions on best fit curves with regressions in the 0.5 range or populations of ~10-20 people. I understand the desire to move to a statistics based approach in medicine, but someone should teach medical researchers statistics. I've worked with engineers that have never had a stats course and they punch data into Excel. Get a curve fit with a ever no slight correlation and get all excited.

Compared to my boss who makes us explain every single outlier point, why it happened, and if possible collect new data if we can fix what went wrong.

That this study was stating the obvious was also noted 16 years ago. Unfortunately, often these follow up comments are very hard to find. Seeing all these comments, the article perhaps should have been pulled.

There's a great ancient method for estimating curves that we used to use all the time in instrumental analysis.

take a strip of paper that has a graph on it

cut out two pieces

the area under the curve that you want to measure

a rectangle a certain amount of units high and wide

weigh each piece of paper

multiply the height and width (in the units you are measuring) of the rectangular piece

divide that by the weight of the rectangular piece

multiply that by the weight of the curve piece

You now have the area under the curve!

It's a lot quicker and easier than most other methods for estimating the area if you are dealing with a complex curve. Of course now that computers are used to gather the data instead of strip charts it's even easier for the computer to just add up the magnitude of all the data points and multiply by some constant to get a decent estimate.

So how do you estimate the error in your calculation due to differing density/thickness/weight throughout the paper? Do you cut up the paper into a thousand identical pieces and weigh each and determine the standard deviation? And then do you cut up multiple identical graph strips (and their inverses) to determine the errors in accuracy and precision in your scissors?

Yeah, pretty much. You'd be surprised at how accurate the method is, modern paper is actually remarkably uniform in composition so your error ends up lying mostly in your cutting technique.

It's not a perfect method but it ends up beating the pants off of most other methods of measuring the area under the curve, especially in how quick and easy it is to perform.

While boat-builders use Simpson's rule on hull surfaces to estimate the displacement...with a slide rule and a sharp pencil.

Oh, but they're trained in Union apprenticeship programs and so could not *possibly* be as bright or talented or well-trained as a Doctor who went to University. And see? This Doctor has a publication! He must deserve 10X the salary of a boat builder.

Okay, I realize you're probably just trolling here, but you do realize that he reinvented integration, not just learned how to solve a couple of integrals, right?

It says something sad about the state of interdisciplinary communication that this was considered worthy of publication, but if you think it reflects poorly on his intelligence, you're missing the point.

Way too much of the practice of being a doctor involves calculus to let that slide.

Almost none of the practice of medicine requires calculus. Trust me, I'm a doctor. There's a lot of use for calculus in medical research, and in deeper understanding of physiology - but it has no bearing whatsoever on my daily work.

"Reinvented" is putting it a bit strongly, at least from the abstract of the paper (I, shockingly, don't have access to the Diabetes Care journal to see the full extent of the "discovery"). As well as I can gather, he noticed the area of a curve can be approximated by making a bunch of rectangles underneath it, and that you can be "clever" and add a triangle above the rectangles to get an even better answer. That's not even close to reinventing integration. To be honest, it's not even integration in a formal sense; no idea of limits seems to be used, for instance, or boundedness, infinite sums, or infimums/supremums.

Did he, say, find the fundamental theorem of calculus and derivatives, along with a few formulae like the binomial theorem which gives the usual power rule? Is he able to compute some integrals symbolically? If so, I'd be impressed. But, and without being able to read the article itself, he seems like a guy who got tired of counting cells on graph paper and noticed he could do a little better by drawing trapezoids.

This isn't integration. This is a numeric technique for estimating the area under the curve (the trapezoidal rule). This is a somewhat different branch of mathematics to integral calculus, which deals in the infinitesimal limits to provide exact results. You can't use integral calculus here, as there is no formula to integrate, only experimental results.

It looks like this area is indeed in need of some interdisciplinary communication: what they really need is for a statistician to come up with a robust form

You subscribe to the common (and completely erroneous) delusion that doctors make a lot of money. While sure it might sound great to say your income is 400k a year as a specialist, and completely ignore the 10+ years of school it took to get there, the student loans, and since medicine is not really a career you can work your way through, that's 10 years of no income too. THEN give half of it to the government in taxes. THEN give half of THAT to the insurance companies for liability insurance. THEN pay for all your supplies. And then you can afford a modest lifestyle.

Boo hoo - I spent 7 years after college getting a PhD in Physics and then did a post-doc after that - and I don't know of many Physicists making 400k a year or even half that, I spend my time desiging x-ray machines that doctors use and I have spent quite a bit of time watching them use the machines - as far as I can tell they are glorified technicians, they do the same type of procedure every day, which mostly involves manual dexterity, they don't have a clue how their equipment works, and on several occas

I don't think it's possible for you to be paying $200k in taxes with an income of $400k and deductions of $100k for insurance premiums and another positive amount for supplies. Assuming $50k in supplies, and living in California with a 9.3% state income tax, your total income tax burden (including self-employed SS/Medicare) is about $110k, not even close to $200k. That would make take-home, after-tax pay $140k. If you live in Florida with no state income tax, your take home pay is about $165k. If you can't get rich off of that over the course of your career, you are doing it wrong, simple as that. Marry someone who is better at handling money than you.

Maybe your doctor friends are so rich that you have lost track of what "modest lifestyle" means to most people vs you?

"If you can't get rich off of that over the course of your career, you are doing it wrong" -- remember, this discussion was started by an article in which a med school graduate and research scientist reinvented the trapezoid method of integration, presumably because he never learned it in math class. So we're not talking math geniuses here.

Composite construction material composed of cement (commonly Portland cement) and other cementitious materials such as fly ash and slag cement, aggregate (generally a coarse aggregate made of gravels or crushed rocks such as limestone, or granite, plus a fine aggregate such as sand), water, and chemical admixtures.

First, does anyone have a link to the actual article? TFS only seems to include an abstract. Second, this was published in 1994. Third, while it may simply seem that the author is rediscovering integration, the field of numerical integration is actually a rather rich one. It's all well and good to say "take an antiderivate and evaluate at the endpoints", but for a function that is found experimentally this is essentially nonsense. While the submitter here claims that this article is simply rediscovering the trapezoid rule, there's actually no such evidence given in the Abstract--algorithms for determining how big of rectangles/trapezoids/etc to use in your calculations is actually an active area of research (albeit usually for the multidimensional case) and it is possible that this researcher did actually discover a better algorithm for deciding how to do the numerical approximations.

Tai's article was printed in February of 1994. An author comment printed in the October 1994 issue is titled "Tai's formula is the trapezoidal rule." [nih.gov] I don't have full text access to either, but the title of the followup is not encouraging.

Actually there appears to be no less than three follow-up commentaries to that article in the same issue.

Apart from the one you mentioned there's R Bender, "Determination of the area under a curve." and T M Wolever, "Comments on Tai's mathematic model.".
In my experience, an article has to be pretty damn bad to get any kind of commentary against it, but three? That basically means it's just as crazy as we think it is.

And sure, numerical integration is a rich field, but real advances in numerical integration aren't published in "Diabetes Care".
Doesn't have to be a math journal, physics or comp sci could be just as plausible, but a medical journal? Not really.

Sure, it's math that has been known by math and physics types for centuries, but what is truly impressive is that a medical researcher, in other words someone who, if they still remember any math is chemical math or statistical math oriented actually managed to handle a topic such as this.What I think is most odd about this is that no-one in his peer review group noticed that this is actually relatively trivial calculus. My nephew has recently applied to study medicine in the university and I was more than

Apparently most slashdotters do math on a daily basis. I can't recall the last time I needed to do integrals - in fact, if you had asked me 5 minutes ago how to calculate the area under a curve, I would have needed a trip to google/wolfram to look it up.

Can't really fault someone who isn't doing it on a daily basis for not knowing the "obvious" answer.

"Apparently most slashdotters do math on a daily basis. I can't recall the last time I needed to do integrals - in fact, if you had asked me 5 minutes ago how to calculate the area under a curve, I would have needed a trip to google/wolfram to look it up."

I haven't done any calculus in XY years but I guarantee you if someone asked "how do I figure out the area under a curve" I'd eventually answer "Calculus", at least before I wrote a medical journal about it and submit it for peer review. I mean he quot

There is a great short story by Jorge Luis Borges, called "Pierre Menard, Author of Don Quixote," wherein the titular character sets out of to write Don Quixote. The fact that Don Quixote was written by Miguel de Cervantes centuries ago is irrelevant. Pierre Menard does not try to copy Cervantes' work, and in fact he avoids reading it to make sure that it does not affect his own authorship. Instead, Menard goes out and makes it so that his combined life experiences inspire him to write a creative work, pulled out of his own imagination, that just so happens to conform, word-for-word, to the original text of Don Quixote. He is not the first to write it, but neither is he plagiarizing. He completes his masterpiece shortly before his death, and it goes largely unnoticed....

The story goes into a critical review of the piece and claims that due to the author's particular circumstances, it is artistically superior to the original Don Quixote.

Concur. It is one of a number of devastating critiques by Borges of the various foibles of literary criticism itself - all told as very short delightful stories. "Pierre Menard" attacks the idea that examining the life of the author is necessary to evaluate a literary work -- that the work itself cannot stand on its own. He destroys the opposite extreme of literary criticism -- essentially the whole approach of deconstructionism - in "The Library of Babel" in which interpretations are read into works independent of any intended meaning of the authors (the books in the story are simply random combinations of symbols), and this was written in the late 1940s, 20 years before "deconstruction" was coined. Taken together he is defending the idea that books actually convey meaning themselves that a reader can apprehend.

And "Tlon, Uqbar, Orbus Tertius" is possible the most idea-dense work in the history of literature, it is a short story that plays with more concepts (with striking effect) than most "novels of ideas" (at the end of the 20th century the New York Times picked it as the greatest short story of the century). I am amused that the Wikipedia entry on the story (last time I checked) is longer than the story itself, but still fails to do justice to all the ideas presented.

Borges was easily the greatest writer of the 20th Century never to receive a Nobel Prize, and I would argue the greatest writer of the 20th Century, period.

Even if he isn't, the failure is on the journal for not properly reviewing the paper. If it's purportedly a mathematical paper (as in, the title starts with, "A Mathematical Model for....") then perhaps a mathematician should look at it.

I don't know what kind of academic curriculum a student could choose these days that would permit them to pursue a career in medical research without ever having learned basic calculus at SOME point. I mean, when I was in high school, having taken AP Calculus AB was more or less a requirement for applying to almost any reasonably competitive four-year university. How do you enter a pre-med program without even knowing what an integral or derivative is? It seems completely implausible to me, given how competitive these programs have become. Moreover, that this author somehow thought it novel to estimate the area under a curve via trapezoidal approximation is not nearly as bewildering as the fact that they should have had the basic research skills to find that their "discovery" amounted to something that is regularly taught to high school kids. To me, that's the real scandal--that someone who can write a journal article doesn't know or care to look for prior research.

And it becomes really, really scary when you realize that this is the level of calculus applied to life-saving techniques in medical science. It can probably explain a lot of medical failures made every year...

I don't think many medial mishaps would be corrected by a better understanding of calculus.

See now, it may actually be that a better understanding of calculus would result in more medical mishaps. Here's why:

Although I think getting surgeons to simply count the number of utensils on the bench before and after each operation would help quite a lot.

It may just be a fluke, but I and several of my classmates observed when we got to high school calc that the higher we got in math the more basic arithmetic and counting errors we made. If this phenomenon holds beyond our ridiculously small self-selected sample (which is a BIG if) then the medical profession may be doing us a big favor by keeping their calculus skills dull, thus keeping the

Of course, I can't read them, because they're behind a paywall. The rights to the paper are owned by the American Diabetes Association, which supports something called the "Washington DC Principles for Free Access to Science" [dcprinciples.org]. This is a lobbying group against free access to scientific publications. They've been fighting open publication since 1994. Here's their latest output, opposition to the Federal Research Public Access Act, which would force all Government-funded research papers onto public servers.

Philosopher: 3 is prime, 5 is prime, 7 is prime, therefore by induction all odd numbers are prime.Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is experimental error, 11 is prime...Computer Scientist: 3 is prime, 3 is prime, 3 is prime, 3 is prime, 3 is prime...Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime...Statistician: In the same of odd numbers: 3, 5, 11, 13, and 29 they are all prime so all odd number are prime.Artist: 1 is prime, 2 is prime, 3 is prime, 4 is prime...

To be fair, I don't think high school courses usually cover numerical approximations to integration. At least here in the UK, our equivalent courses cover analytical integration of continuous functions in one variable, with just a brief covering of the principles behind integration (using the rectangular approximation, IIRC, along with the notion that as the width of the rectangles approach zero the error introduced disappears). But only the analytical approach is actually tested, so I wouldn't be surpris

I agree. That points out a anecdote that happened just this evening with my 6 year old son. Understand, we are one of those 'crazy' home school families, so, yes, it will seem a little bizarre. Anyway, we were playing "Matter", a solid/liquid/gas trivia game with our son. He got the question "when you freeze water, it's weight A) get lighter, B) stays the same C) gets heavier.

When our son was clearly guessing at the answer, we we simply walked through it. It went like this:

Dad: What is water made of?
Son: Hydrogen and Oxygen.
Dad: What is Hydrogen and Oxygen made of?
Son: Atoms?
Dad: What makes atoms weigh something?
Son: Gravity.
Dad: What is gravity?
Son: The force that pulls matter together.
Dad: OK, what happens what are you doing to the ice when you melt it?
Son: Making it hotter.
Dad: So, what happens to the atoms?
Son: The move faster?
Dad: And?
Son: They take up more space?
Dad: And?
Son: B, its weight stays the same!

This is not how math and science are normally taught. Normally, the same information is taught as "If you freeze water it's weight doesn't change. Remember that." If your lucky it is "If you freeze matter, its weight doesn't change. Remember that."

Yes, we could have just had him memorize the trivia, but instead we helped him "Rediscover" that mass doesn't change weight when you heat it.
The fact that a public school would just have him memorize the fact is one of the reasons we home school.

1) The man names the method after himself. I can see the smug look on his face when he figured out how to integrate, and decided to name his newfound discovery after himself. That's a big no no in science.
2) It's been cited 137 times since it was published. Most recently in June. That means that there has been ~137 people that cited it without seeing that it's just an integral.
3) It completely reaffirms the whole stereotype of the premedical student memorizing everything they need to get into medicine but understanding nothing.

Actually, from the abstract this looks like a moderately interesting paper. Also note that the slashdot summary is (as often the case) wrong. You can't solve the problem the paper is referring to with integral calculus.

The curve that the paper is talking about is an experimental result, not a formula. All you have are the experimental samples from the curve. Without a formula, you CAN'T do integration, and must rely on a numerical technique. What he's 'invented' here is the trapezoidal rule. He'd do even better with something like Simpson's rule, but that might be impossible to apply if the sample points are not evenly spaced. Similar problems occur for the various Runge-Kutta methods.

Although the numerical technique that claims to be invented here is indeed a basic numerical technique, the paper is interesting for pointing out that the even cruder numerical techniques that have been used before are overestimating the curve area, and that is an interesting result.

You may laught at this, but you find the same thing in all fields. Programming language designers are writing papers on decades old language features, user interface researchers are getting lots of citations for decades old ideas or gimmicks from scifi movies, and theoretical computer science authors are woefully ignorant of statistics and machine learning. Mathematicians and physicists aren't immune either.

My sister, a librarian, was laughing when relating a story of software engineer explaining to them the concept of meta-data with respect to a library collection. He acted as if this was a concept well beyond their grasp. She finally moved the discussion along by saying "You mean it's like a card catalog, and the records are like the cards in the card catalog?"

As a physics grad student, I TA a LOT of life-science, pre-med students for introductory physics. In these courses, calculus is not necessary. Considering how horrific an average student performs when confronted a problem requiring more than 3 lines of algebra manipulations, I would not be surprised if there's a statistic somewhere more than half of MDs cannot do first-year college level math. I also tutored people taking the MCAT, again, calculus not necessary.

That's OK, when I was a grad student in Molecular and Cell Biology, we of course had to TA the 100 level intro course which, of course, was on the pre med track. The faculty was on this kick that college students could not express themselves so they decided that all of the tests were to be exposition style. Sentences and paragraphs and the like.

We hated that. As it turned out, the faculty's supposition was correct. The majority of students could not write a simple declaratory sentence, much less a coherent paragraph. Grading them was a nightmare, especially the premeds who would cry and moan over 1 or 2 points. Try as we might, I doubt that we taught them a whole lot (either English or Molecular Biology)

Then at least some of them went to Medical School.

But medicine these days is a really a long, drawn out vocational school. There is very little 'Science' and even less 'Humanity'. It is memorize and practice. To a large degree this is unavoidable - there is a huge volume of baseline knowledge to acquire in a relatively short period of time. But given that the premedical experience is likewise short on science and humanities, your average physician really does not have the broad educational experience that many folks assume they do.

This isn't as stupid as it sounds, because up to the 1980s spectrometers and chromatographs had pen-and-paper plotters, not personal computers for data recording. Numerical integration would've been a waste of time without a computer.

Though a very valid comment (Simpson's Rule would be better), note that you may not be able to apply Simpson's Rule here directly. The basic form of Simpson's Rule needs evenly spaced sample points, which might not be the case for experimental results.

Wait, what?... When did integration require you to have a 'formula' for the function?...

Or rather to put it in another way; a data set as in the measurements from a lab test do translate into a function (for the points where we have data) and if we decide on how to interpolate between values we have a function which is continuous. So yeah, the slashdot item is spot on and you're probably in the same category as dr. Tai.

To apply the rule for a polynomial term - "add one to the exponent of x, then divide by the new exponent",

Of course if you're talking about a numerical approximation to an integral it's different. But that isn't what rve said.

What rve said is irrelevant.

Before that rule existed, before the Fundamental Theorem of Calculus existed, "Tai's Method" was the way integration was done. And of course "Tai's Method" taken to the limit of zero-width trapezoids was fundamental to proving the Fundamental Theorem of Calculus.

Of course with non-zero width trapezoids it is merely an approximation... for a continuous function. For a function defined by discreet data points, and assuming you're linearly interpolating between data points, then this is as good as it gets.

Either way, the point is, this is anything but new or novel. It is how integrals were calculated literally hundreds of years ago, and it was never forgotten, at least not by anyone who took and remembers Calc I.

The story is one of the problem of overspecialisation. This is a very good example, because it's a very basic principle in mathematics that someone sufficiently advanced in the field of medicine to be publishing research papers. It's a problem all over academia, however. Pick up a journal from a distantly related field and you'll be pretty much guaranteed to see a paper inventing or discovering something that everyone in your field has known about for decades.

.... this sounds so familiar... in the 1990's, one group inside Siemens discovered that contacts made of little carbon blocks can be used in CT scanners to transfer current and data from x-ray tube and detector (part of gantry that is moving around patient) to stationary part of gantry/scanner.

After proudly presenting that at internal meeting, one guy said: ".... but we have been using it for decades in trains.... for the same purpose..."

I should add that it's a very difficult problem to solve. In general, people need a lot of specialised knowledge to make a valuable contribution to a specific field. Acquiring the same level of knowledge of multiple fields would take many years. That said, it's always worth spending time with people outside your own discipline. Richard Hamming, for example, claimed that he always had lunch with the physicists or chemists in his group, rather than with other mathematicians, and often provided or gained n